Two Aging Mathematicians

Two aging mathematics professors, Al Gebra and Cal Culus were sitting at their favourite Thai restaurant, Phun Kshuns, reminiscing about when they first met, while their waiter, Theo Rumsanrools (who also took Professor Gebra's Advanced Mathematics course at Cart Eshian University) eavesdropped on their conversation as he brought them their meals.

"You know what occurred to me the other day, Cal?" asked Professor Gebra.
"What's that?" said Professor Culus.
"Well, when we met, the square of your age contained the same three digits as the square of my age, just in a different order," Professor Gebra continued.
"This will really blow your socks off then, Al! If you take the square of the sum of our ages when we met and split it into two 2-digit numbers, you'd have my age then and your age now!" Professor Culus exclaimed.

Wanting to impress his professor, Theo interjected.
"I thought you were much younger than that, Professor Gebra," Theo said.
Said Professor Gebra, "And just how old am I then, young fellow?"

Answer

"Well, you're 81, sir," Theo replied.
"Right you are! And how did you figure that out so quickly?" Professor Gebra asked.
"You said the square of your ages when you and Professor Culus met had only three digits, so the oldest you could have been was 31. That means that there were only a few possible ages the two of you could have been. Those being (with the associated squares):
14 & 31 (196 & 961),
13 & 31 (169 & 961),
16 & 25 (256 & 625),
12 & 21 (144 & 441),
13 & 14 (169 & 196).

Now the square of the sum of each of these pair of ages are respectively:
2025,
1936,
1681,
1089,
729.

And from those figures I concluded that Professor Culus must have been 16 when you met and hence you, Professor Gebra, are now 81!" Theo explained.
"Aren't you a bright young fellow! You should come and study my Advanced Mathematics course at Cart Eshian University!" Professor Gebra exclaimed.
"Uh, I already do," Theo said sheepishly.Hide